Diffusers and rear-wheel wakes

There is a persistent notion amongst some Formula One technical analysts that the low pressure wake behind the rear wheels can be connected to the lateral extremities of the diffuser airflow, thereby enhancing the flow capacity of the underbody, and its downforce-generating potential. In particular, the notion has been repeatedly promoted by Autosport Technical Consultant Gary Anderson:

"Mercedes has worked very hard in making the low pressure area behind the rear tyres connect up to the trailing edge of the diffuser. In effect this gives the diffuser more extraction capacity," ( The key technical developments from Australia, 25th March 2017).

Now, it's certainly true that if the diffuser is expanded in a lateral direction without causing separation of the boundary layer, then the expansion ratio of the diffuser will be increased, and it'll generate more downforce. It's also true that the wakes shed by the wheels are areas of low pressure, situated as they are behind rotating bluff bodies. So surely, one might think, there will be a pressure gradient directed towards those rear-wheel wakes, and surely the airflow exiting the diffuser can be connected to them, thereby increasing its effective expansion ratio?

Unfortunately, whilst the wakes behind bluff bodies do indeed tend to be regions of low pressure, they are also regions of high turbulence, and the airflow 'sees' a region of turbulence as an obstruction. Directing the lateral extremities of diffuser airflow towards the rear wheel wakes does not therefore offer a straightforward boost in the power of the diffuser, and could even promote diffuser separation.

One illuminating way to understand this is to look at the Reynolds-averaged Navier-Stokes (RANS) equations for a flow-field containing turbulence. A solution of these equations represents the mean velocity flow field $\overline{u}$ and the mean pressure field $\overline{p}$ in a region of space. For a time-independent incompressible flow, each component $\overline{u}_i$ of the mean velocity vector field is required to satisfy the equation

$$
\rho (\mathbf{\overline{u}} \cdot \nabla) \overline{u}_i  = - \frac{\partial \overline{p}}{\partial x_i} +\frac{\overline{\tau}_{ij}}{\partial x_j} - \rho \frac{\overline{u'_i u'_j}}{\partial x_j} \;.
$$ This equation is simply a version of Newton's second law, $F=ma$, albeit with the accelerative term on the left-hand side, and the force terms on the right-hand side.

In the case of a continuous medium, the density $\rho$ is substituted in place of the mass, and $(\mathbf{\overline{u}} \cdot \nabla) \overline{u}_i$ represents the acceleration experienced by parcels of air as the velocity field changes from one spatial position to another. 

Each term on the right-hand side of the equation represents a different type of force. The first term $- \partial \overline{p}/\partial x_i$ is the familiar pressure gradient. The negative sign indicates that the force points in the opposite direction to the gradient: the fluid will be pushed away from high pressure, and sucked toward low pressure.

Pressure, however, is only the isotropic component of stress. When the isotropic component has been subtracted from the total stress, what remains is called the 'deviatoric' stress $\tau_{ij}$. This represents the stresses which occur due to viscosity $\nu$. These are the forces which occur within a continuous medium when there are shear motions. In the case of a Newtonian fluid such as air, the deviatoric stress is a function of the viscosity and the velocity shear:

$$ \tau_{ij} = \rho \nu \bigg[ \frac{\partial u_i}{\partial x_j}+ \frac{\partial u_j}{\partial x_i} \bigg] $$In general, forces are generated by spatial gradients of the stress, and the second term on the right-hand side of the RANS equation represents the force due to the spatial gradient in the mean deviatoric stress. These 'tangential' forces are crucial inside the boundary layer of a fluid, but more generally they play a role wherever one layer of fluid runs parallel to another layer travelling at a different speed. Here, the viscosity entails that momentum is transferred from the higher velocity layer to the lower velocity layer, helping to pull it along. This is a source of acceleration in the flow-field which cannot be explained by pressure gradients alone.

The third term on the right-hand side of the RANS equation represents the effective force due to spatial gradients in the turbulence. In a turbulent flow-field, the velocity at a point is decomposed into a sum $u_i = \overline{u}_i + u'_i$ of the mean-flow $\overline{u}_i $ and the turbulent fluctuations, $u'_i$. The expression $\overline{u'_i u'_j}$ represents a type of turbulent stress, hence its spatial gradient provides another source of acceleration in the mean flow-field.

This third term is crucial to understand why the rear wheel wakes behave like obstructions in the flow-field. Note the negative sign associated with the turbulent-stress term. That entails that the force vector points away from a region of turbulence. Airflow exiting a region of low turbulent intensity will effectively experience a repulsion force as it approaches a region of high turbulence.  

Hence, trying to join the diffuser-flow to the rear wheel wake is not necessarily a good idea. A better idea is to create vortices from the edges of the diffuser which push the rear wheel wake further outboard. This might enable one to increase the expansion ratio of the diffuser without provoking separation.